SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 335, 1025-1028 (1998)

Previous Section Next Section Title Page Table of Contents

2. The Zeeman splitting in CCH

Experimental studies have established beyond doubt that CCH is a linear molecule and has a [FORMULA] ground state (Graham et al. 1974; Sastry et al. 1981; Gottlieb et al. 1983). From both laboratory and astrophysical data it has been established that the angular momentum coupling scheme of CCH follows Hund's case [FORMULA] (Townes & Schawlow 1975) to a reasonable approximation (Sastry et al. 1981; Ziurys et al. 1982; Gottlieb et al. 1983). Accordingly, we have

[EQUATION]

where [FORMULA] is the rotational angular momentum, [FORMULA] the electron spin, [FORMULA] the angular momentum and [FORMULA] the spin of the hydrogen nucleus. Yet, as indicated by Ziurys et al. (1982), due to the mixing of J-states by the hyperfine interaction, J is no longer a good quantum number, and the usual selection rules [FORMULA] no longer hold. Rather, the selection rules for electric dipole transitions now are

[EQUATION]

Fig. 1 shows the allowed transitions between rotational levels [FORMULA] and N, with fine and hyperfine splittings included.

[FIGURE] Fig. 1. The [FORMULA] energy level diagram of CCH showing all allowed transitions. (Mixing of the J energy levels causes the usual selection rule [FORMULA] to break down, hence the transition [FORMULA].) When [FORMULA], the lower rotational level has only one fine structure J level, viz. [FORMULA].

The magnetic fields prevailing in the interstellar medium are so weak that they do not decouple the angular momentum [FORMULA] and the nuclear spin [FORMULA]. Their effect is just to lift the degeneracy of the hyperfine levels F by splitting them into [FORMULA] levels, labeled by the quantum number [FORMULA] ([FORMULA]). The Zeeman interaction energy is usually written as

[EQUATION]

where [FORMULA] is the Bohr magneton, B the magnetic field, and [FORMULA] a dimensionless coefficient known as the Landé factor. The frequency of the emitted radiation for a transition between Zeeman sublevels [FORMULA] and [FORMULA] is given by

[EQUATION]

where [FORMULA] is the frequency of the transition without magnetic field, [FORMULA], and the index F corresponds to the lower level.

The selection rules for transitions between the sublevels of the rotational levels [FORMULA] and N are

[EQUATION]

From the expressions given by Gordy & Cook (1970) it is easily established that, for the values of the rotational levels we are considering in this paper ([FORMULA]), the Landé factor can be written as

[EQUATION]

Under interstellar conditions, [FORMULA] is much less than the full line width at half-maximum (FWHM), so that the individual transitions cannot be resolved, making the Zeeman effect very difficult to detect. However, among all these transitions, the so-called [FORMULA] transitions (corresponding to [FORMULA]) are elliptically polarized, with the circularly polarized portion due to the line-of-sight component of the magnetic field. By substracting the left from the right circular polarization of the radio signal, the amplitude of the obtained Stokes V spectrum is proportional to the strength of the line-of-sight component of the magnetic field and to the average frequency separation between the oppositely polarized [FORMULA] components (Verschuur 1969; Troland & Heiles 1982). The average is obtained by weighing the frequency separation of the individual pairs of [FORMULA] transitions by their relative intensity. The relevant expressions for the latter are (Gordy & Cook 1970)

[EQUATION]

where F is the smaller of the two F's involved in the transition and the upper sign is taken throughout for the [FORMULA] transition and the lower sign for the [FORMULA] transition. The coefficients P and Q are independent of [FORMULA].

Tables 1 to 5 give the resulting average frequency separation of the left and right circularly polarized portions of the [FORMULA] transitions for the transitions listed in Ziurys et al. (1982).


[TABLE]

Table 1. Separation, [FORMULA], of the Zeeman [FORMULA]-components for the [FORMULA] transition of CCH



[TABLE]

Table 2. Separation, [FORMULA], of the Zeeman [FORMULA]-components for the [FORMULA] transition of CCH



[TABLE]

Table 3. Separation, [FORMULA], of the Zeeman [FORMULA]-components for the [FORMULA] transition of CCH



[TABLE]

Table 4. Separation, [FORMULA], of the Zeeman [FORMULA]-components for the [FORMULA] transition of CCH



[TABLE]

Table 5. Separation, [FORMULA], of the Zeeman [FORMULA]-components for the [FORMULA] transition of CCH


Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
helpdesk.link@springer.de